Flow and Fluid Behaviour of theMantle - Quantitative Plate Tectonics - page 348

applicability of this method, here we will follow
the more rigorous approach of Spiegel and Vero-
nis ( 1960 )andFurbish( 1997 ). Let us assume that
pressure, p , temperature, T , and density, ¡,have
expressions of the type:

where 4 ¡ 0 is the maximum change of the static
density variation ¡ 0 over the distance H .The
quantity © should be considered as the maxi-
mum acceptable error in simplifying the equa-
tions governing mantle convection, in particular
thermal convection. Therefore, we shall neglect
any term in these equations having the same
order of magnitude of ©. Condition ( 13.102 )is
sufficient to produce simplified equations when
the velocities have infinitesimal magnitude, for
example at the onset of convection. However,
in the context of non-linear dynamics it is also
necessary to assume that the relative magnitude
of the fluctuations associated with fluid motion
does not exceed the static variation ©.
Thus, we require that:
LJ LJ LJ LJ

p.r;t/ D p C p 0 . z / C p 0 .r;t/
T.r;t/ D T C T 0 . z / C T 0 .r;t/
¡.r;t/ D ¡ C ¡ 0 . z / C ¡ 0 .r;t/

(13.98)

where p, T ,and¡ are global spatial averages,
p 0 , T 0 ,and ¡ 0 are variations about the mean in
the pure conductive limit (that is, in hydrostatic
conditions), and the primed variables denote fluc-
tuations associated with fluid motion.
It is also assumed that these variables are
defined in a fluid layer of thickness H , heated
from below and maintained at temperature T m ,
while the top is cooled from above and main-
tained at a lower temperature T a .Nowletus
introduce scale heights for the state variables
defined above. These parameters measure the dis-
tance over which the corresponding state variable
changes by a factor e (the Napier's constant)
along a vertical profile:

LJ LJ LJ LJ O.©/

¡ 0
¡

(13.103)

In normal conditions, we expect that ¡ 0
¡ 0 , thereby it should not be necessary to verify
a posteriori that condition ( 13.103 ) is satisfied.
Now let us consider the equation of state ( 13.19 )
of the fluid, which can be written in the form:

LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ

¡ D ¡.p;T/ (13.104)

1 1

1
p

dp 0
d z

1
T

dT 0
d z

H p D I H T D I

Expa n di ng ¡ in Taylor series about the state

LJ LJ LJ LJ LJ LJ LJ LJ

¡ ;p;T yields:

1

1
¡

d¡ 0
d z

H D

(13.99)

@T T T C

@ ¡ @ ¡

@p .p p/

¡ D ¡ C

Our first approximation is to assume that H
throughout the fluid is much less than the smallest
scale height, and that the latter coincides with the
density scale height:

@T 2 T T 2

@ 2 ¡

@T@p T T .p p/

@ 2 ¡

1
2

1
2

C C

@ 2 ¡
@p 2 .p p/ 2

1
2

C C :::

H<<H ¡ (13.100)

(13.105)

This condition can be rewritten as follows:
LJ LJ LJ LJ

where it is intended that th e d erivatives are calcu-
lated at the mean state p;T .Nowletusdefi ne
the average coefficient of thermal expansio n , ',
and the average isothermal compressibility, “,as
follows:

LJ LJ LJ LJ

1
¡

d¡ 0
d z

<< 1
H

(13.101)

Then, integrating from z D 0to z D H we
conclude that:

LJ LJ LJ LJ T DT;pDp I “ LJ LJ LJ LJ T DT;pDp

(13.106)

1
¡

@¡
@T

1
¡

@¡
@p

'

¡ 0
¡ ©<<1

(13.102)